# Z Transforms

## Difference Equations

A difference equation is a discrete equivalent of a differential equation, used in situations where only discrete values can be measured:

becomes

These can be solved numerically by just evaluating the output for each value of n. For example:

This evaluates to:

Alternatively, there is an analytical solution...

## The z Transform

Consider a discrete sequence . The z transform of this sequence is defined as:

A closed-form expression can generally be found by the sum of the infinite series. For example, the z transform of the unit step :

This is a geometric series with , , hence the sum is

Taking a z transform of a difference equation converts it to a continuous function. The z domain is similar to the laplace domain, but for discrete time signals instead.

## z Transform Properties

z transforms have linearity, the same as laplace and fourier transforms.

### First Shift Theorem

If is a sequence and it's transform, then

For example, if :

For :

### Second Shift Theorem

The function is defined:

Where is the unit step function. The function , where is a positive integer, represents a shift to the right of this function by sample intervals. If this shifted function is sampled, we have . The second shift theorem states:

## Inverse z Transforms

z transforms are inverted using lookup tables, but to get them into a recognisable form, some manipulation is often needed, including partial fractions. For example, finding the inverse transform of :

The first term can be seen immediately from the table:

The second term rearranges to give:

This is in the form of the second shift theorem, so this can be applied to give:

Thus,

## Example

Solve , where , .

Taking z transforms:

Rearranging and using initial conditions:

Using partial fractions:

Using inverse transforms straight from the table to get the solution: